A227799 -id:A227799 - cp's OEIS frontend

A227797 Number of composites removed in each step in the Sieve of Eratosthenes for 10^8.

49999999, 16666666, 6666666, 3809523, 2077920, 1598400, 1128284, 950133, 743581, 564099, 509508, 413103, 362709, 337382, 301484, 261684, 230683, 219393, 196552, 182782, 175351, 159910, 150351, 138581, 125778, 119552, 116075, 110630, 107564, 102739, 90485

Offset: 1

Eric F. O'Brien, Jul 31 2013

The number of composites <= 10^8 for which the n-th prime is the least prime factor.
pi(sqrt(10^8)) = the number of terms of A227797.
The sum of a(n) for n = 1..1229 = A000720(10^8) + A065855(10^8).

			For n = 3, prime(n) = 5, a(n) = 6666666: 5 divides 10^8 20000000 times. 10 is the least common multiple of 2 (prime(1)) and 5 and 15 is the least common multiple of 3 (prime(2)) and 5; thus [10^8 / 10] multiples of 5 and [10^8 / 15] multiples of 5 have already been eliminated by a(1) and a(2), and thereby respectively reduce a(3) by 10000000 and 6666666 offset by [10^8 / 30] multiples of 5 which would otherwise excessively reduce a(3) by 3333333 because 30 is the least common multiple of 2, 3 and 5. a(3) is further reduced by 1 as 5 itself is not eliminated.

Eric F. O'Brien, Table of n, a(n) for n = 1..1229

Cf. A133228, A145538-A145540, A227155, A227798, A227799, A145532-A145537.

Writing floor(a/b) as [a / b]:
a(1) = [10^8 / 2] - 1.
a(2) = [10^8 / 3] - [10^8 / 6] - 1.
a(3) = [10^8 / 5] - [10^8 / 10] - [10^8 / 15] + [10^8 / 30] - 1.
a(4) = [10^8 / 7] - [10^8 / 14] - [10^8 / 21] - [10^8 / 35] + [10^8 / 42] + [10^8 / 70] + [10^8 / 105] - [10^8 / 210] - 1.

A227798 Number of composites removed in each step of the Sieve of Eratosthenes for 10^9.

Original entry on oeis.org

499999999, 166666666, 66666666, 38095237, 20779220, 15984016, 11282834, 9501331, 7435826, 5640969, 5095068, 4131143, 3627360, 3374293, 3015292, 2616982, 2306411, 2192860, 1963654, 1825278, 1750219, 1595163, 1499127, 1381337, 1253379, 1191536

Offset: 1

Eric F. O'Brien, Jul 31 2013

a(n) = the number of composites <= 10^9 for which the n-th prime is the least prime factor.
pi(sqrt(10^9)) = the number of terms of this sequence.
The sum of a(n) for n = 1..3401 = A000720(10^9) + A065855(10^9).

			a(1) = 10^9 \ 2 - 1.
a(2) = 10^9 \ 3 - 10^9 \ (2*3) - 1
a(3) = 10^9 \ 5 - 10^9 \ (2*5) - 10^9 \ (3*5) + 10^9 \ (2*3*5) - 1
a(4) = 10^9 \ 7 - 10^9 \ (2*7) - 10^9 \ (3*7) - 10^9 \ (5*7) + 10^9 \ (2*3*7) + 10^9 \ (2*5*7) + 10^9 \ (3*5*7) - 10^9 \ (2*3*5*7) - 1.

Eric F. O'Brien, Table of n, a(n) for n = 1..3401

Cf. A133228, A145538-A145540, A227155, A227797, A227799, A145532-A145537.

cp's OEIS Frontend

A227797 Number of composites removed in each step in the Sieve of Eratosthenes for 10^8.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula